Uniform asymptotics for orthogonal polynomials with exponential weights-the riemann-hilbert approach

Let Q(x) = q _2mX ^2m + q _2m-1x ^2m-1 + ⋯be a polynomial of degree 2m with q _2m > 0, and let {π _n(x)} _n≥1 be the sequence of monic polynomials orthogonal with respect to the weight w(x) = e ^-Q(x) on ℝ. Furthermore, let α _n and β _n denote the Mhaskar-Rakhmanov-Saff (MRS) numbers associated with Q(x). By using the Riemann-Hilbert approach, an asymptotic expansion is constructed for π _n(c _nz + d _n), which holds uniformly for all z bounded away from (-∞, -1), where c _n = 1/2(β _n - α _n) and d _n = 1/2(β _n + α _n). © 2005 by the Massachusetts Institute of Technology.